In order to determine the axion relic density today, the adiabatic approximation is again appropriate after \(T_2\) . The comoving number of axions \(N_a^{\prime }\)  is constant during this final period (and different from that during the first oscillation period \(N_a\) ). Using \(q_{\rm kin}\)  in Eq. (REF ) and the mean axion velocity at
\(T_{\rm QCD}\)  in Eq Mean theta dot, \(N_a^{\prime }\)  is well approximated by [1]}
\(N^{\prime }_a= \frac{\rho _{a,2} a_2^3}{m_a}=2\,\frac{m_a f_a^2 a_2^3}{{\cal N}^2}= \frac{\dot{\theta }_{\rm tr}}{{\cal N}}\,f_a^2\,a_{\rm QCD}^3 \,.\) 
